= Solution
The <cubic lattice> has <graph vertices> $\mathbb Z^d$ and an <edge> between $x,y$ when $\sum_i|x_i-y_i|=1$. In independent <site percolation>, every <graph vertex> is open with <probability> $p$ and closed with <probability> $1-p$, independently. The open subgraph contains only open <graph vertices> and the <edges> between them. Let $C_p(0)$ be the open <connected component of a graph> containing $0$, with $C_p(0)=\varnothing$ when $0$ is closed. The <percolation probability> is
$$
\boxed{\theta(p)=\mathbb P_p(|C_p(0)|=\infty)=\mathbb P_p(0\leftrightarrow\infty)}.
$$
Since the <cubic lattice> is a <locally finite graph>, the <König infinity lemma> identifies an infinite open <connected component of a graph> with the existence of an infinite open <graph ray> from the origin. In particular, the origin itself must be open. The <percolation critical probability> is $p_c=\inf\{p:\theta(p)>0\}$.
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